The following is taken from Introduction to Topological Manifolds by John Lee, is the following observations correct. $\varphi = \tilde{q_2}$ is a homeomorphism, since it is bijective by the fact that $\tilde{q_1} \circ \tilde{q_2} = \text{Id}_{Y_1}$. To see why that's true, suppose $\varphi$ wasn't surjective, then the composition won't even be defined, and if $\varphi$ wasn't injective then there would exist $y_1 \neq y_2 \in Y_1$, such that $q_2(y_1) = q_2(y_2) = y \in Y_2$, and we'd have $q_1(y) = q_1(q_2(y_2)) = q_1(q_2(y_1))$, implying that $y_1 = y_2$ a contradiction, thus proving bijectivity of $\varphi$. It follows that $\tilde{q_1}$ is a continuous inverse for $\tilde{q_2}$ proving that $\varphi$ is a homeomorphism.
Have I left out any details filling in the argument? Have I filled in unnecessary details?
A homeomorphism $h: X \to Y$ is a continuous function such that there is another continuous function $h': Y \to X$ such that $h' \circ h = 1_X$, where $1_X$ is the identity function on $X$ and $h \circ h' = 1_Y$ as well. So this $h'$ is the set-theoretic inverse of $h$ and must also be continuous.
The proof of Lee shows exactly that $\tilde{q}_1$ and $\tilde{q}_2$ are each other's continuous inverse.
It's true that a function between sets has an inverse iff it is injective and surjective, but that's irrelevant to this proof. We just direct;y show the inverses property by uniqueness.